In my heart of hearts
Hamlet:
Give me that man
That is not passion's slave, and I will wear him
In my heart's core, ay, in my heart of heart,
As I do thee.
The Bard was a logical man, and he went about coining sensible phrases in a rational fashion. Thus, Hamlet does not say "in my heart of hearts," but "in my heart of heart"—that is, at the "heart" (center) of my heart. The phrase is in fact a synonym for "In my heart's core." And like the heart of an artichoke, the heart of Hamlet's heart is its most tender part. He reserves this region of his affection for men who aren't slaves to their passion, who are governed by reason, like his friend Horatio (whom he addresses here) and, indeed, like the phrase-coining Shakespeare.
We've perverted the phrase into "in my heart of hearts" by way of expressions like Ecclesiastes' "vanity of vanities." But where Ecclesiastes had a number of vanities from which to elect a chief or encompassing vanity—presumption—one doesn't have a number of hearts. Even granting that we use "heart" mostly as a metaphor and not with reference to the organ, we never mean to speak of having more than one.